When they reach the outside world, and see the sun rise for the first time, the sun rises from bottom right to top left. The sun would only do this in the southern hemisphere. The film is clearly set in and around a post apocalyptic Washington DC. Interestingly magnetic north could change, but the direction in which the sun rises, is physically incapable of change.
About 5 minutes into the movie, Logan and Francis 7 are on the escalator to Carrousel. Logan says "Why were you so angry just then? What did he do to you?" then Francis 7 says "That's red's going to run, I can always tell." Neither of their mouths are moving.
When Logan and Jessica first emerge from the ice cave and see the sun, they have no idea what the sun is. Later, when they attempt to re-enter the city at the water falls, Jessica asks what they're for. Logan says that he thinks it's part of the city's power generation system. He says "The ocean tides are turned into energy somehow". If they no longer have any knowledge of what a "sun" is, it's doubtful they know what an "ocean" is, and especially doubtful they understand the concept of "tides".
Jenny Gutter, while talking to Ustinov about his burial reassures him that (his death) is a long way off. However the character had never even seen an old person so would have no idea about a lifespan. She could not have simply been intending to be polite because the city dwellers had no cliches or conventions about natural death.
Logan and Jessica first see the sun as it rises from behind some distinctive mountain peaks. At the end of their first day, the sun sets behind the very same peaks where it rose. The sunrise seems to be the sunset footage, reversed, which would explain why the sun rises in the wrong direction for Washington, DC.
When the dome city is destroyed and the Old Man sees the people emerging, he is standing by a small campfire. When the people surround him, several people are standing where the camp fire was originally burning.
The cut made by the laser on Logan's shirt runs straight up and down from shoulder to nipple. Later, after destroying Box the cut in the shirt runs from the base of the neck to his armpit at about a 45 degree angle.
When Logan and Francis are chasing the runner in the open-air lobby, the runner climbs up one floor and starts to clamber over the railing. As shown from below, the support column next to the place where he's climbing up is about flush with the railing. Then the shot changes and the runner is shown still clambering over the railing. However, now the support column extends a few feet back from the railing, and there's a large potted plant next to it that wasn't there before.
When Logan and Jessica swim back into the city and pull themselves up out of the water, Jessica's left palm does not have a life-clock on it. Nor does the palm of Logan's left hand. In the beginning scenes with the babies Logan holds his left hand to the glass and we see the red crystal on his palm for the first time.
When the City's computer asks Logan to sit down, the large circular light directly above the interrogation chair is not lit. When it cuts to a shot behind Logan facing the screen, the lamp is on, but when it cuts back to Logan, the lamp is just lighting up.
When Logan and Jessica are in the ice cave with Box, during one scene Logan is talking to Box while it appears Box is circling around to come back to face Logan. While Box is moving, his upper torso is obviously moving and slightly swaying and when Box tries to come to a full stop in front of Logan you can see that Box appears slightly off balance before they cut the shot. In the next scene we only see a frontal shot of Box by himself and he is perfectly still.
Logan's Sandman uniform top is a form-fitting body shirt with a gray collar and a matching tunic that fits over this with a gray band across the chest. After Logan and Jessica remove their clothes and get under the furs to keep warm, Logan only puts on the body shirt with the gray collar and discards the tunic. When they're shown shortly thereafter in a long shot at the top of a waterfall, however, Logan is wearing the tunic with the gray band across the chest (which disappears thereafter).
Throughout most of the movie, Logan's uniform is black (possibly dark navy blue), but when he and Jessica are challenged by the people with the light-spears, it is a bright sky-blue. It might have been an effect from the bright light that was shined on him, but nothing else looked different.
When Logan and Francis terminate the first runner near the start of the movie and the "clean up crew" comes to spray/disintegrate the body, the shadow of the wires that hold up the disintegration unit are visible on the wall.
After Logan and Jessica enter Washington, they are seen wading in the reflecting pool on the National Mall, approaching the Washington Monument from the west, with the Capitol visible behind. That would put the Lincoln Memorial behind them. Yet the next scene has them climbing the steps of the Lincoln Memorial they would have just passed.
When Logan and Jessica emerge from the ice cave, they see the sun rising from behind a great mountain. As they make their way to Washington DC, they travel through valleys and mountains to get there. There are no mountains such as these or any other kind even within miles of the vicinity of Washington DC.
The last act established the city as being on the coast (presumably the Atlantic Ocean.) Yet after Logan and Jessica flee the city and travel across the countryside, they approach Washington DC from the west, crossing the Potomac and traveling up the Mall from the Lincoln Memorial to the Capitol.
When meeting the other Sandman that had been to Cathedral and fought the Cubs, Logan told him to get a new face, so he wouldn't be recognized, later after Logan goes to Cathedral he also goes to New You to change his face. There is no reason for a Sandman to change his face to protect his identity, as they can be identified by their uniform.
The city's calendar is apparently based on astrological signs rather than the standard months we use today. The film takes place during the "month" of Capricorn, which is during December and January of a standard year, when it would be bitterly cold outdoors in the Washington DC area. However, the outdoor weather in the film is obviously very mild. (It is possible, however, that the astrological "months" are used during different times of year, or that some type of massive catastrophe occurred that changed the climate.)
There is a shot looking down the steps of the Lincoln Memorial showing Logan and Jessica climbing the steps. At the right is a column which has been composited into the shot. Less than two minutes later they are shown climbing the steps of the Capitol building. The cut in Logan's shirt has reversed sides because this is a flopped repeat of the earlier shot but without the column.
The City's Computer commands 'Logan 5' to approach and identify before he is identified, and calls him 'Logan 5' one more time before he identifies, thus calling him by his identity twice before he is identified. Francis had to place his palm on the identifier to be identified before Logan went in.
Logan has a tear in his black shirt between his left nipple and left armpit. When they reach the grand hall and are climbing a staircase a long shot of him and Jessica show the tear on the opposite side - it looks as though the film was reversed for that shot (it goes back to the left side where it was originally from then on).
When the Doc switches off the healing function and the lasers begin cutting Logan for the first time, it is clear that the sparks are a superimposed visual effect - they are cut off for the whole upper part of the frame.
When Logan and Jessica are descending the stairs at the terraced water complex, the top of a tree is visible in the distance; yet in the previous wider shot, only terraced structures (a matte drawing) and the ocean (superimposed video) were visible from that angle.